The scope of simple interest is most often short-term transactions (with a period of up to one year) with a single interest calculation (short-term loans, bill credits) and less often long-term transactions.
For short-term transactions, the so-called intermediate interest rate, which is understood as the annual interest rate reduced to the investment term Money. Mathematically, the intermediate interest rate is equal to the percentage of the annual interest rate. The formula for accruing simple interest using an intermediate interest rate is as follows:
FV=PV(1+f*r),
FV = PV (1 + t * r / T),
t -- the term of investment of funds (in this case, the day of investment and the day of withdrawal of funds are taken as one day); T is the estimated number of days in a year.
For long-term transactions, the calculation of simple interest is calculated by the formula:
FV=PV(1+r*n),
where n is the period of investment of funds (in years). ,
The scope of compound interest is long-term transactions (with a period exceeding a year), including those involving intra-annual interest.
In the first case, the usual compound interest formula is applied:
FV = PV (1 + r)n.
In the second case, the compound interest formula is applied, taking into account intra-annual accrual. Intra-annual interest is the payment of interest income more than once a year. Depending on the number of income payments per year (m), the intra-annual accrual can be:
The accrual formula for semi-annual, quarterly, monthly and daily compound interest is as follows:
FV = PV (1 + r / m)nm,
where PV is the original amount;
g -- annual interest rate;
n is the number of years;
m -- the number of intra-annual accruals;
FV -- accumulated amount.
Interest income in the case of continuous interest calculation is calculated according to the following formula:
where: e \u003d 2, 718281 is a transcendental number (Euler number);
e?n is the incremental factor, which is used both for integer and fractional values of n;
Special designation of the interest rate for continuous interest calculation (continuous interest rate, “growth force”);
n is the number of years.
With the same amount of the initial amount, the same term of investment of funds and the value of the interest rate, the amount returned is greater in the case of using the intra-annual accrual formula than in the case of using the usual compound interest formula:
FV = PV (1 + r / m)nm> FV = PV (1 + r)n.
If the income received when using intra-annual accruals is expressed as a percentage, then the interest rate received will be higher than that used in the usual calculation of compound interest.
Thus, the initially declared annual interest rate for calculating compound interest, called nominal, does not reflect the real efficiency of the transaction. The interest rate that reflects the actual income received is called effective. The classification of interest rates for intra-annual compound interest is clearly illustrated in the figure.
The nominal interest rate is set initially. For each nominal interest rate and based on it, the effective interest rate (re) can be calculated.
From the formula for accruing compound interest, you can get the formula for the effective interest rate:
FV = PV (1 + r)n;
(1 + re) = FV / PV.
Here is the formula for accruing compound interest with intra-annual accruals, in which r / m percent is accrued every year:
FV = PV (1 + r / m)nm.
Then the effective interest rate is found by the formula:
(1 + re) = (1 + r/m)m,
re = (l + r/m)m- 1,
where re is the effective interest rate; r -- nominal interest rate; m -- the number of intra-annual payments.
The value of the effective interest rate depends on the number of intra-annual accruals (m):
The area of simultaneous application of simple and compound interest are long-term operations, the term of which is a fractional number of years. In this case, interest can be calculated in two ways:
In the first case, the compound interest formula is used for calculations, in which there is an exponentiation to a fractional power:
FV = PV (1 + r)n+f,
where f is the fractional part of the investment period.
In the second case, the so-called mixed scheme is used for calculations, which includes a compound interest formula with an integer number of years and a simple interest formula for short-term transactions:
FV = PV (1 + r)n * (1 + f * r),
FV = PV (1 + r)n * (1 + t * r / T) .
1. Compound interest
1.1. Compound annual interest
If interest is not paid immediately after it is accrued, but is added to the amount of the debt, compound interest is applied. The addition of accrued interest to the amount of the accrual base is called interest capitalization.
Let's use the same notation as in the accrual formula for simple interest. At the end of the first year, the interest is equal to Pi, and the accrued amount will be Р + Рi = Р(1 + i).
By the end of the second year, it will reach the value
P(1 + i) + P(1 + i)i = P(1 + i)2 etc. At the end of the nth year, the accumulated amount will be equal to
S = P(1 + i) n (1.1)
Interest for this period:
I \u003d S - P \u003d P [(1 + i) n - 1]
The value (1 + i)n is called the compound interest multiplier. The values of this factor for integers n are given in the tables of compound interest.
Compound rate accumulation time is usually measured as ACT/ACT.
If the interest rate changes in the contract, then apply the formula:
S= P (1+ i 1)n1 (1 + i2)n2 … (1+ik)nk ,
Where i1, i 2, … i k - consecutive values of rates; n1,n2,…,nk – periods for corresponding rates.
Often for interest calculation, the term is not an integer.
There are three methods for calculating interest.
1The accrued amount is calculated using the formula:
S= P (1+ i 1)na (1 + i2)nb,
Where na is the integer part of the accrual period, nb is the fractional part of the accrual period.
1. Assumes the calculation of interest for an integer number of years using the compound interest formula and for the fractional part of the term using the simple interest formula:
S = P(1+ i 1)na (1+ nb i)
2. In the rules of a number of commercial banks, for some operations, interest is charged only for a whole number of years or other periods of accrual.
The fractional part of the period is discarded:
S= P (1+ i 1)na
In order to compare the results of accumulation at different interest rates, it is enough to compare the corresponding accumulation multipliers. At the same levels of interest rates, the ratios of these multipliers depend significantly on the term. For n > 1, as the term increases, the difference in simple and compound interest increases. The ratio of the growth multipliers is shown in fig. 3.
Rice. 3. Ratio of accumulation multipliers for simple and compound interest
1.2 Doubling formulas
Based on formulas for simple and compound interest
S= P + I= P + Pni = P(1+ni),
S= P (1+ i)n
we get the following doubling formulas:
- doubling simple interest:
2= 1 + ni -> n = 1/I ,
n \u003d ln 2 / ln (1 + i) \u003d o.69315 / ln (1 + i).
In general, to increase the initial amount by N times:
- for simple interest:
N= 1+ ni -> n = N-1/ I ,
Doubling compound interest:
N= (1+i)n -> ln N / ln (1+i) .
When working with compound interest, the rule of 72 is applied: if the interest rate is i, then the doubling of capital occurs in about 72 / i years.
For example, at a rate of 12%, a doubling of capital occurs after 6 years.
1.3. Accrual of interest m times a year. Nominal and effective rates
In modern conditions, interest is capitalized, as a rule, not once, but several times a year - for six months, quarters, etc. Some foreign commercial banks they even practice daily calculation of interest. Let the annual rate be equal to j, the number of periods of accrual in a year is m. Each time interest is charged at the rate j/m. Rate j is called nominal. Growing formula:
S = P(1+ J/m)mn , (1.2)
Where N= nm is the total number of accrual periods.
The real, or effective, interest rate is the annual compound interest rate, which gives the same result as m, which is a one-time charge of interest at the rate j/m. It measures the real relative income that is received in the whole year.
Denote effective rate through i. Accrual multipliers calculated at effective and nominal rates must be equal to each other:
(1 + i)n = (1 + j/m)mn .
From here
I = (1 + j/m)m – 1.
The effective rate for m > 1 is greater than the nominal one.
Determination of the nominal rate j for the given values of i and m:
1.4. Compound Rate Discounting
Let's determine the initial amount by the accumulated amount through mathematical discounting:
P = S / (1 + I) n
and when interest is compounded m times a year:
P = S / (1 + J/m) mn
In banking accounting, a complex discount rate is used. In these cases, the discounting process occurs with a slowdown, since each time the discount rate is applied not to the initial amount, but to the amount discounted at the previous time step. Discounting at a complex discount rate is more beneficial for the debtor than at a simple discount rate:
P = S(1 – d)n ,
where d is the compound annual discount rate.
1.5. Nominal and effective discount rates
Discounting can be done not once, but m times a year, i.e. each time accounting is done at the f/m rate. In this case
P = S (1 – f/ m) mn ,
where f is the nominal discount rate.
The effective discount rate (d) characterizes the degree of discounting for the year. We define it based on the equality of discount factors:
(1 – d) n = (1 – f / m)mn ,
Where
d = 1 - (1 - f / m)m .
The effective discount rate in all cases where m > 1 is less than the nominal one.
2. Inflation
2.1 The concept of inflation
Inflation as an economic phenomenon has existed for a long time. It is believed that it appeared almost with the emergence of money, with the functioning of which is inextricably linked.
The term inflation (from the Latin inflatio - swelling) was first used in North America during the Civil War of 1861-1865. and denoted the process of swelling of paper money circulation. In the 19th century the term is also used in England and France. The concept of inflation became widespread in the economic literature in the 20th century. after the First World War, and in Soviet economic literature from the mid-1920s.
The most general, traditional definition of inflation is the overflow of circulation channels with money supply in excess of the needs of trade, which causes depreciation. monetary unit and a corresponding increase in commodity prices.
However, this definition of inflation cannot be considered complete. Inflation, although it manifests itself in the rise in commodity prices, cannot be reduced to a purely monetary phenomenon. This is a complex socio-economic phenomenon generated by disproportions in reproduction in various fields market economy. Inflation is one of the most pressing problems modern development economies in many countries around the world.
Regardless of the state of the monetary sphere, commodity prices may increase due to changes in the dynamics of labor productivity, cyclical and seasonal fluctuations, structural shifts in the reproduction system, market monopolization, state regulation of the economy, the introduction of new tax rates, devaluation and revaluation of the monetary unit, changes in market conditions, the impact foreign economic relations, natural disasters, etc. Therefore, the rise in prices is caused by various reasons. But not every rise in prices is inflation, and among the reasons for rising prices mentioned above, it is important to single out truly inflationary ones.
First of all, it should be noted that the rise in prices may be associated with an excess of demand over supply of goods. However, such an increase in prices associated with a disproportion between supply and demand in a particular commodity market is not yet inflation. Inflation is an increase in the general level of prices in the country, which occurs due to a long-term disequilibrium in most markets in favor of demand. In other words, inflation is an imbalance between aggregate demand and aggregate supply.
Inflation is manifested primarily in the depreciation of money in relation to gold, goods and foreign currencies. As a result, the gold content of the national currency decreases, so the price of gold rises.
Almost all countries face inflation, and last years characterized by an increase in its rate. We can say that the world has become more inflationary.
Separate aspects of inflation are described by such concepts as “disinflation”, “deflation”, “stagflation”. Disinflation means a slowdown in the rate of inflation. Deflation is a long-term decline in the price level. The term “stagflation” is derived from stagnation and inflation and means high inflation with slow or zero growth in real output. Often this term is used to characterize inflation with a simultaneous decline in output.
2.2 Causes of inflation
There are many causes of inflation, however, each country has its own socio-economic conditions for its occurrence. Allocate external and internal causes of inflation.
External causes include:
1. Internationalization of economic relations: the presence of inflation in other countries affects the dynamics of domestic commodity prices through the prices of imported goods. The country's central bank to create its own foreign exchange reserves buys foreign currency from commercial banks, issuing for these purposes an additional national currency which increases the amount of money in circulation.
2. World economic crises. So, the global structural crisis of the 70s. of the 20th century led to an increase in prices for Natural resources 7 times, including crude oil - 20 times. As a result, prices for finished products jumped sharply in Japan, the US, Western Europe. This factor is of great importance, for example, for Belarus, whose economy is 90% or more dependent on the import of fuel and energy resources. Rising prices for them is one of the main reasons for the unwinding of the inflationary spiral.
Internal causes are due to the state of the economy of a given country. Among them are:
First. State budget deficit. If it's covered by loans Central Bank country, the amount of money in circulation rises sharply, but it is not supported by the release of goods, which leads to inflation.
Second. military spending. First, they increase the expenditure side of the budget, being a constant cause of the budget deficit, which, as noted, leads to inflation. Secondly, people employed in the military sector of the economy do not create a consumer product, but act in the consumer market only as buyers, increasing effective demand. Consequently, military appropriations are a powerful factor in inflation, as they cause a huge increase in money supply without appropriate product coverage.
Third. Spending for social purposes is not adequate to efficiency national economy. In cases of economic crises, a decline in production, the standard of living of the population decreases. The government seeks to support the population through additional allocations for social purposes (wage indexation, payment of various benefits, including unemployment, various additional payments, etc.), which leads to an increase in the amount of cash in circulation and increases inflation.
Fourth. Inflationary expectations, which are one of the main factors of inflation. When inflation starts, the population plans its behavior in anticipation of further price increases. It begins to acquire goods in excess of its current needs. There is a “flight from money”. Demand begins to stimulate supply, which pushes prices up. In addition, inflation expectations are included in long-term contracts (usually at least a year), salaries and other payments. High wages driven by prior expectations stimulate further price increases. It blocks the government's efforts to reduce inflation.
Fifth. Excessive investment in certain sectors of the economy, for example, in Agriculture not giving due economic effect.
Sixth. Structural disturbances in the economy - disproportions between accumulation and consumption, supply and demand, government revenues and expenditures, and other factors.
2.3 Types of inflation
In the world economic theory In practice, two types of demand inflation and supply inflation are known.
Demand-pull inflation results from an increase in aggregate demand under full load conditions production capacity, and hence the impossibility of responding with an increase in output (Fig. 29). The reasons for the increase in demand may be; increase in government orders and growth wages and an increase in the purchasing power of the population. A lot of money appears in circulation, not backed by goods.
Supply-side (cost) inflation occurs as a result of rising prices due to an increase in production costs. The reasons for the increase in costs can be - an increase in prices for raw materials, the actions of trade unions to increase wages, monopolistic or oligopolistic pricing of resources, etc.
2.4 Types of inflation
Inflation is distinguished depending on the pace, the nature of the course, expectations and the extent of coverage.
In terms of inflation, we can distinguish:
- moderate inflation (price growth is less than 10% per year);
-galloping inflation (price growth ranges from 10 to 200% per year);
- hyperinflation (price growth is more than 50% per month).
The most detrimental to the economy is hyperinflation, which is expressed in an astronomical increase in the amount of money in circulation. The role of money in the economy is greatly reduced, and industrial enterprises switch to other forms of settlements (for example, barter, mutual settlements).
On the basis of anticipation, one can single out expected inflation, which is expected and predicted by the government and the population, and unexpected inflation, which is characterized by a sudden jump in prices. The latter has an ambiguous effect on the behavior of the population, depending on the state of inflation expectations. If there are no inflationary expectations in the country, then the population, counting on the short-term rise in prices, acquires less and saves more money. Demand decreases and puts pressure on producers, inducing them to reduce prices (the effect of Pigou's law is manifested). Macroeconomic balance is being restored. If inflation expectations are high in the country, a sudden rise in prices encourages the population to buy goods for the future. Demand is growing, which leads to further price increases and higher inflation.
By the scale of coverage, one can single out local inflation, which takes place in individual countries, and global inflation, covering a group of countries or entire regions.
According to the nature of the flow, open inflation is distinguished, characterized by a long rise in prices, and suppressed, arising from firm “frozen” retail prices for goods and services with a simultaneous increase cash income population. In this case, goods disappear from the shelves and become scarce, and prices rise on the “black market”.
Open inflation is inherent in countries with market economy, where the free interaction of supply and demand contributes to an open, unrestricted rise in prices as a result of a fall in the purchasing power of the monetary unit.
Although open inflation and distorts market processes, nevertheless, it retains the role of prices as signals showing producers and buyers areas for profitable investment of capital. Thus, open inflation itself acts as a kind of anti-inflationary tool.
Suppressed inflation is inherent in an economy with administrative control over prices and incomes. It is called "repressed" because the tight control over prices and incomes does not allow inflation to openly manifest itself in the only form available to it: in the growth of money prices. In such a situation, inflation takes on an “underground” character, prices are outwardly stable, but since the money supply has actually increased, the excess money is transformed into a shortage of goods, which cannot be compensated by an increase in production. With suppressed inflation, only a part of banknotes is money, while the other, non-commodified part, immediately turns into false money, while no one knows what he has at his disposal - money or false money? This mystery affects the behavior of buyers and sellers in different ways.
Buyers try to “catch” a scarce product by turning banknotes into real money. But it is precisely the scarcity of goods that means that the purchase becomes an accident, a fortune, a lottery. There are queues - constant, dull and embittered. Sellers begin to speculate in scarce goods. A "black market" appears - an illegal form of inflation in the conditions of its suppression.
The "Black Market", to some extent, shows the true prices of goods. At the same time, it turns out that buyers are robbed twice: administratively fixed prices hypocritically testify to their “stability” (and, therefore, there is no reason to raise wages!), but people who receive income at the level of official price tags of empty stores actually have to buy goods at black market prices. Moreover, the illusion of price invariance creates the appearance of economic prosperity, misleading buyers, sellers, and the government (until now, part of our society sighs at those “low” and “stable” prices that did not reflect any economic reality).
Suppressed inflation is incurable, it can only be “anesthetized” by driving even deeper, not allowing it to manifest itself, and thereby “exploding” the entire economy. Yes, and this can be achieved only by administrative methods. As a result, the economy is in for a real catastrophe. The fact is that the suppression of inflation for decades distorts prices so much that real economic processes are simply not signified, society lives by self-deception and is accustomed to it.
Inflation is measured using a price index. In practice, the gross national product index, the wholesale price index and the index consumer prices.
- The index of the gross national product, called the GNP deflator (GDP), expresses the ratio of the volume of GDP in actual prices to the volume of the same GDP in the so-called basic prices, most often in prices previous year.
- Wholesale price indices are relative indicators that characterize the relationship of prices over time (usually the prices of the base year are taken as 100, and the prices of subsequent years are recalculated in relation to the base year). For example, the average price of gasoline in the base year 1995 was 54 thousand rubles. per ton, and in 1996 it was already 162 thousand rubles, then the gasoline price index will be equal to 300% (162 thousand: 54 thousand) x 100%. That is, the average price in reporting year in relation to the base increased by 3 times.
When calculating inflation using the consumer price index (CPI), the starting point is the “consumer basket” – a set of goods and services purchased by an average urban dweller during a given period of time (year, quarter, month). The cost of the basket for the previous year, quarter, month is taken as a base, starting point and compared with the cost of the basket, calculated in the prices of this month, quarter or year. The PPI is calculated using the Laspeyres index.
2.4 Determinants of inflation
At this stage, there is complete agreement among scientists about the determinants of the inflationary process, but there is no explicit agreement on the results of the impact on the inflationary process. In order to understand the determinants of inflation and the sources of disagreement between different scientific schools, it is worth considering the following equation:
P = MV/Y, (2.1)
where P = price level, M = money supply in the economy, V = money turnover rate in the economy, Y = real output in the economy. The money supply turnover rate measures how often money circulates in the economy and the volume of transactions that is created. So if 1 EEK created 3 EEK in the volume of transactions, its turnover is 3. It is also worth noting that if the value of the money supply is determined by a specific indicator, then the turnover should be calculated to reflect a specific situation. Let us rewrite the previous equation (2.1) in terms of changing parameters, where d represents the change.
dP = (dM) (dV) / (dY)
The left side of the equation is the inflation rate, and the right side shows the three determinants of the inflation rate.
A) Change in money supply
If the volume rises with other parameters constant, then the inflation rate will rise. This is the basis for the arguments of monetarist theorists who believe that there is no relationship between real output and the money supply, and that the turnover rate is stable for a long time and “loose” monetarist policy (increasing the money supply) is the cause of high inflation. Although some admit that monetary policy may have a short-term effect on real output, most argue that there is no long-term effect. There is also an opinion that although turnover may change over time, these changes appear after a long period of time and are unlikely to have a significant effect on inflation.
b) Change in money supply turnover
If the turnover increases with the remaining parameters constant, then the inflation rate will increase. Economists have long argued why the circulation of the money supply changes over time. One of the determining factors is technological progress. It changes the way money is saved and the way people spend money, thus influencing the turnover of money. In hyperinflation, people are unwilling to hold cash amounts of money and prefer to purchase real goods. The reluctance to accumulate money leads to an acceleration in the turnover of money. Thus, if the central bank rapidly increases the money supply, this invariably leads to an increase in the rate of inflation.
c) Change in real output
If the volume increases with other parameters constant, then the inflation rate will decrease. Often this is the main argument of the Keynesians for easing monetary policy during economic downturns. They argue that an increase in the money supply leads to a concomitant increase in real output and inflationary processes are imperceptible or do not exist.
2.5 Consequences of inflation and anti-inflation policy
The economic and social consequences of inflation are complex and varied. Its small pace contributes to the growth of prices and the rate of profit, being, thus, a factor in the temporary revival of the economic situation. As inflation deepens, it turns into an obstacle to reproduction, exacerbates economic and social tensions in society.
Galloping inflation disorganizes the economy, causing damage to both large corporations and small businesses, primarily due to the uncertainty of market conditions. Inflation makes it difficult to conduct effective macroeconomic policy. In addition, uneven price growth increases disproportions between sectors of the economy and distorts the structure of consumer demand. The price ceases to fulfill its main function in a market economy - to be an objective information signal.
Inflation activates the flight from money to goods, turning this process into an avalanche, exacerbates the hunger for goods, undermines incentives for money accumulation, disrupts the functioning of monetary system, revives barter.
High growth rates of the general price level also have a negative impact on the fiscal system – tax revenues depreciate. So, if taxes are charged, for example, in the III quarter, and paid in the IV quarter of the year, then with hyperinflation, the real value of tax revenues to the budget falls.
In conditions of inflation, the savings of the population are depreciated, the losses are borne by banks and institutions that provide loans. The internationalization of production facilitates the transfer of inflation from country to country, complicating international monetary and credit relations.
Inflation also has social consequences, it leads to a redistribution of national income, it is like a supertax on the population, which causes the growth rate of nominal and real wages to lag behind the sharply rising prices for goods and services. All categories of hired workers, freelancers, pensioners, rentiers suffer from inflation, whose incomes either decrease or increase at a rate less than the rate of inflation.
The negative social and economic consequences of inflation are forcing governments of different countries to take this phenomenon into account in their economic policy. At the same time, first of all, economists are trying to find an answer to such an important question - to eliminate inflation through radical measures or adapt to it. This problem in different countries is solved taking into account their specifics. In the USA and England, for example, the task of combating inflation is set at the state level. Other countries are developing a set of adaptation measures (indexation, etc.).
anti-inflation policy.
In the anti-inflationary policy of states, two approaches can be distinguished. The first approach (it is being developed by representatives of modern Keynesianism) provides for an active budget policy - maneuvering public spending and taxes for the purpose of influencing the payment-capable demand.
With inflationary, excess demand, the state limits its spending and raises taxes. As a result, demand decreases and inflation rates decrease. However, at the same time, the growth of production is also limited, which can lead to stagnation and even crisis phenomena in the economy, to an increase in unemployment. This is the cost to society of curbing inflation.
budget policy It is also carried out to expand demand in a recession. If demand is insufficient, programs of public investment and other spending are carried out, and taxes are lowered. Low taxes are established primarily for people with medium and low incomes, who usually quickly use (spend) their income. It is believed that this increases the demand for consumer goods and services. However, stimulating demand budget funds may increase inflation. In addition, large budget deficits limit the government's ability to maneuver taxes and spending.
The second approach is recommended by neoclassical economists, who highlight monetary regulation, which indirectly and flexibly affects economic situation. This type of regulation is central bank(formally uncontrolled by the government), which changes the amount of money in circulation and rates loan interest thus affecting the country's economy. Economists of the neoclassical direction believe that the state should carry out deflationary measures to limit effective demand, since stimulation economic growth and artificial maintenance of employment by reducing natural level unemployment leads to loss of control over inflation.
So
etc.................
Bespalova Ekaterina
The content of the work corresponds to the stated topic and is presented in accordance with a well-planned plan. In the "Introduction" section, the topic, goals and objectives of the work are defined, as well as research methods are listed. The goals and objectives of the work are quite competently and convincingly confirmed by the materials of the work. The authors successfully used such methods as analysis, synthesis, comparison. The materials of the work indicate that the researchers carefully studied the theoretical material on this topic, carried out calculations and made their own conclusions. The applied value of this topic is very high and affects the financial, economic, demographic and other spheres of our life. An understanding of percentages and the ability to perform percentage calculations and calculations is necessary for every person, since we encounter percentages in Everyday life. In the theoretical part design work everything you need to know about simple and compound interest is presented: formulas, explanations and calculations using these formulas. A good addition to the work is the research part, which is devoted to a comparative analysis of compound and simple interest, which shows the suitability of compound interest in banking system. The student independently conducted a study on deposits individuals in various banks, making a reasonable conclusion that compound interest plays a large role in the economy and the banking system. The material may be useful to teachers of mathematics, economics, students of educational organizations.
State budget professional educational institution Republic of Khakassia "Tekhnikum public utilities and service"
Project theme:
« The use of compound interest in economic calculations
Scientific adviser: Cherdyntseva L.A.
Student: Bespalova Ekaterina Andreevna
Group: TT-11
Abakan, 2016
Introduction
Every day we do the same thing - we live, work, eat and sleep, for us it is everyday life. We don't even notice that many terms are related to everyday life. For example, the economy is part of everyday life. People take part in daily economic activity, live in economic environment. In turn, no economy can do without interest. Interest is all around us.
But interest appeared in ancient times among the Babylonians. Cash settlements with percentages were common in ancient Rome. The Romans called interest the money that the debtor paid to the lender for every hundred. From the Romans, interest passed to other peoples.
At present, interest is applied in all economic spheres activities: in enterprises, in statistics, in the banking system, etc. We will show our work on the example of banks.
Why banks? Banks are in the center economic life serve the interests of manufacturers by linking cash flow industry and trade, agriculture and population. All over the world, banks have significant power and influence, they manage the huge money capital flowing to them from enterprises and firms, from merchants and farmers, from the state and private individuals.
Why does a person bring his savings to the bank? Of course, to ensure their safety, and most importantly - to receive income. And here, knowledge of the formula for simple or compound interest, as well as the ability to make a preliminary calculation of interest on a deposit, will come in handy more than ever. After all, forecasting interest on deposits or interest on loans is one of the components of a reasonable management of your finances.
This is the relevance of the topic.
Objective:
Study of simple and compound interest in economic calculations.
Tasks:
Compare simple and compound interest on deposits of individuals.
Compare income on deposits of individuals using compound interest formulas depending on the time period.
Conduct an analysis of income on deposits of individuals in various banks.
Interest
Interest is the amount paid for the use of money.
Interest is divided into simple and compound
1) Simple interest - interest that is charged on the initial amount.
S - the amount of funds due to be returned to the depositor at the end of the deposit (ie deposit).
I - annual interest rate
t - the number of days of accrual of interest on the attracted deposit
K - number of days in a calendar year (365 or 366)
P - the initial amount of funds attracted to the deposit
We came up with a problem so that you can see how simple interest is applied in bank calculations.
Task 1.
The bank made a contribution in the amount of 100,000 rubles, and after 5 years the account had 168,000 rubles. Determine the bank's interest rate using simple interest.
Solution:
I= (168000-100000)*(365*100%)/100000*1825=13.6%
Answer: 13.6% rate.
2) Compound interest - interest earned on accrued interest.
I - annual interest rate;
j is the number of calendar days in the period following which the bank capitalizes accrued interest;
K is the number of days in a calendar year (365 or 366);
P is the initial amount of funds attracted to the deposit;
n - the number of operations for capitalization of accrued interest during general term attracting funds;
S - the amount of funds due to be returned to the depositor at the end of the deposit term. It consists of the amount of the deposit plus interest.
And now we will solve the problem in the same way, but with compound interest
Task 2.
The bank made a deposit of 100,000 rubles. at 13.6%, for 5 years. Interest is charged once a year. How much money will the depositor withdraw from the account at the end of 5 years?
Solution:
S= 100000* (1+ (13.6%*365)/ 365*100%) 5 \u003d 100000 * 1, 1365 \u003d 189187, 2 rubles.
Answer: 189187.2 rubles.
Let's compare simple and compound interest to understand the difference between them:
Task 3. A deposit of 100,000 rubles was made to the bank. at 12% for 10 years. Determine how much money will be through each year, using simple and compound interest.
In the table we see that it is more profitable to use compound interest:
Graph of capital growth using simple and compound interest:
And now let's compare the compound interest on the deposit, depending on the time period.
Task 4. A deposit of 100,000 rubles was made to the bank. for 1 year at an interest rate of 12% per annum. Compare the amounts that will be due back to the depositor when interest is calculated: daily, weekly, monthly, quarterly, semiannually and annually.
In the table we see that the more often the interest calculation interval, the more income we get.
Studying simple and compound interest, we analyzed in which bank it is better to invest money at the moment and why.
We took three banks as a basis - these are Binbank, Alfa-Bank and VTB 24.
VTB 24 - Profitable deposit
Alfa-Bank - Pobeda deposit
Binbank - deposit "Maximum income"
Task 5. We have 500,000 rubles. and choose which bank to put this amount to get the highest income for 1 year.
At the moment, it is best to make a deposit in Alfa-Bank
Conclusion:
Conducted a study of simple and compound interest in economic calculations.
We compared simple and compound interest on deposits of individuals.
We compared the income on deposits of individuals using compound interest formulas depending on the time period.
Conducted an analysis of income on deposits of individuals in various banks
. REFERENCES AND INTERNET RESOURCES
1. Chetyrkin, E. M. financial mathematics/ E. M. Chetyrkin,
textbook. - 6th ed., corrected. - M.: Delo, 2006. - 399 p.2. Samarov, K. L. Financial Mathematics: Prakt. course: study guide / K. L Samarov. - M.: Alfa-M; INFRA-M, 2006. - 78 p.
3. Financial mathematics: a textbook for universities / P. P. Bocharov. - 2nd ed. - M.: Fizmatlit, 2005. - 574 p.
4 Financial mathematics: textbook.-method. complex / S. G. Valeev. - Ulyanovsk: UlGTU, 2005. - 106 p.
5. Financial mathematics. V. Malykhin: http://www.finansmat.ru/.
6. Financial mathematics. A. Fedorov (lectures): http://wdw2005.narod.ru/FM_lec.htm#_Toc179997391.
7. Mathematical Bureau: http://www.matburo.ru/index.php.
8. Financial mathematics (lectures):
http://treadwelltechnologies.com/index.html.
9. The financial analysis: http://www.finances-analysis.ru/financial-maths/.
10. Knowledge - to the masses: http://www.finmath.ru/.
It is a well-known situation that the same amount of money is not equivalent in different periods of time. Accounting for the time factor in financial transactions carried out through the calculation of interest.
Interest money (interest) is the amount of income from lending money in any form (loans, opening deposit accounts, buying bonds, renting equipment, etc.).
The amount of interest money depends on the amount of debt, the term of its payment and the interest rate that characterizes the intensity
interest calculation. The amount of debt with accrued interest is called the accrued amount. The ratio of the accrued amount to the initial amount of debt is called the accrual multiplier (coefficient). The time interval for which interest is calculated is called the accrual period.
Using simple bets interest, the amount of interest money is determined based on the initial amount of the debt, regardless of the number of accrual periods and their duration according to the formula:
The above formula is used to determine the value of the accumulated cost of capital for short-term financial investments.
If the term of the debt is given in days, the following expression must be inserted into the above formula:
where 5 is the duration of the accrual period in days;
The number of days in a year can be taken exactly - 365 or 366 (exact interest) or approximately - 360 days (ordinary interest). The number of days in each whole month during the term of the debt can also be taken exactly or approximately (30 days). In world banking practice, the use of:
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approximate number of days in each whole month and ordinary interest is called "German practice";
the exact number of days in each month and ordinary interest - "French practice";
the exact number of days and exact percentages - "English practice".
Depending on the use of a particular practice of calculating interest, their amount will vary.
Consider examples of financial and economic calculations for securities.
Example 7.1.
Savings certificate with a face value of 200 thousand rubles. issued on 20.01.2005 due on 05.10.2005 at 7.5% per annum.
Determine the amount of accrued interest and the redemption price of the certificate when using various methods of interest calculation.
Let's determine the exact and approximate number of days until the certificate is redeemed.
tT04H = 11 days of January + 28 days of February + 31 days of March + 30 days of April + 31 days of May + 30 days of June + 31 days of July + 31 days of August + 30 days of September + 5 days of October = 258 days.
Iapprox \u003d 11 days of January + 30 x 8 days (February - September) + 5 days of October \u003d 256 days.
Certificates earn income at an interest rate. There are three ways to calculate interest:
1) exact interest, loan term - the exact number of days:
Іfinal \u003d 0.075 x 200 x 258/365 \u003d 10.6 thousand rubles; certificate redemption price:
51 \u003d 200 + 10.6 \u003d 210.6 thousand rubles;
2) ordinary interest, loan term - the exact number of days, the redemption price of the certificate:
52 \u003d 200 + 10.8 \u003d 210.8 thousand rubles;
3) ordinary interest, loan term - an approximate number
Іbіkn = 0.075 х 200 х 256/360 = 10.7 thousand rubles, certificate redemption price:
53 \u003d 200 + 10.7 \u003d 210.7 thousand rubles.
Example 7.2.
The Bank accepts deposits for 3 months at a rate of 4% per annum, for 6 months at a rate of 10% per annum and for a year at a rate of 12% per annum. Determine the amount that the owner of the deposit will receive 50 thousand rubles. in all three cases.
The amount of the deposit with interest will be:
1) for a period of 3 months:
S \u003d 50 x (1 + 0.25 x 0.04) \u003d 50.5 thousand rubles;
2) for a period of 6 months:
S \u003d 50 x (1 + 0.5 x 0.1) \u003d 52.5 thousand rubles;
3) for a period of 1 year:
S \u003d 50 x (1 + 1 x 0.12) \u003d 56 thousand rubles.
When deciding on the placement of funds in a bank, an important factor is the ratio of the interest rate and the inflation rate. The inflation rate shows how many percent prices have increased over the period under review, and is defined as:
The inflation index shows how many times prices have risen over the period under review. The inflation rate and the inflation index for the same period are related by the ratio:
where Ju is the inflation index for the period;
N is the number of periods during the period under consideration.
The inflation rate for the period.
Example 7.3.
Determine the expected annual inflation rate at a monthly inflation rate of 6% and 12%.
Ju = (1 + 0.06)12 = 2.01.
Therefore, the expected annual inflation rate will be = 2.01 - 1 = 1.01, or 101%.
Ju = (1 + 0.12)12 = 3.9.
The expected inflation rate will be:
3.9 - 1 = 2.9, or 290%.
Inflation affects the profitability of financial transactions.
The real value of the accumulated amount with interest for the deadline, given by the time the money is loaned, will be:
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Example 7.4.
The bank accepts deposits for six months at a rate of 9% per annum. Determine the real results of the deposit operation for a deposit of 1000 thousand rubles. with a monthly inflation rate of 8%.
The amount of the deposit with interest will be:
S \u003d 1 x (1 + 0.5 x 0.09) \u003d 1045 thousand rubles.
The inflation index for the term of the deposit is equal to:
Ju = (1 + 0.08)6 = 1.59.
The accumulated amount, taking into account inflation, will correspond to the amount:
1045 / 1.59 \u003d 657 thousand rubles.
When using compound interest rates, interest accrued after the first accrual period, which is part of the total term of the debt, is added to the debt amount. In the second accrual period, interest will accrue based on the original amount of the debt, increased by the amount of interest accrued after the first accrual period, and so on for each subsequent accrual period. If compound interest is accrued at a constant rate and all accrual periods have the same duration, then the accrued amount will be equal to:
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where P is the initial amount of the debt;
in - interest rate in the accrual period;
n is the number of accrual periods during the term.
Example 7.5.
Deposit 50 thousand rubles. deposited in the bank for 3 years with compound interest at the rate of 8% per annum. Determine the amount of accrued interest.
The amount of the deposit with accrued interest will be equal to:
S \u003d 50 x (1 + 0.08) 3 \u003d 63 thousand rubles.
The amount of accrued interest will be:
I \u003d S - P \u003d 63 - 50 \u003d 13 thousand rubles.
If interest were accrued at a simple rate of 8% per annum, their amount would be:
I \u003d 3 x 0.08 x 50 \u003d 12 thousand rubles.
Thus, the calculation of interest at a compound rate gives a large amount of interest money.
Compound interest can be compounded several times a year. At the same time, the annual interest rate, on the basis of which the amount of interest in each accrual period is determined, is called
nominal annual rate percent. With a debt term of n years and compound interest accrual m times a year, the total number of accrual periods will be equal to:
The accumulated amount will be equal to:
1) term of debt:
Example 7.6.
The depositor makes a deposit of 40 thousand rubles. for 2 years at a nominal rate of 40% per annum with monthly accrual and interest capitalization. Determine the accumulated amount and the amount of accrued interest.
The number of accrual periods is equal to:
Therefore, the accumulated amount will be:
A bill of exchange or other monetary obligation before the maturity date on it can be bought by the bank at a price less than the amount that must be paid on them at the end of the term, or, as they say, discounted by the bank. In this case, the bearer of the obligation receives money earlier than the period specified in it, minus income
bank in the form of a discount. The Bank, upon the due date of payment of a bill or other obligation, receives the full amount indicated in it.
If the period from the moment of accounting to the moment of repayment of the obligation will be some part of the year, the discount will be equal.
Brief theoretical background
In medium and long-term financial and commercial transactions, interest may not be paid immediately after they are accrued, but added to the amount of debt. In this case, compound interest is used for accrual.
When accruing compound interest (compound interest) is such a method in which the amount received at the previous stage of accumulation or discounting is taken as the basis for calculating interest. In this case, it is often said that interest is charged on interest.
Unlike simple interest, the base for calculating compound interest does not remain constant, but increases with each step in time. Compound interest accumulation is the successive reinvestment of funds invested at simple interest for one accrual period.
The accrued amount of compound interest is calculated by the formula S=P( 1+r) t, where t – number of accrual periods.
Example 2.1. What value will reach the value of the debt, equal to 1 million rubles. in five years with growth at a compound rate of 15.5% per annum?
S\u003d 1,000,000 (1 + 0.155) 5 \u003d 2,055,464.22 rubles.
Contracts usually specify an annual rate r and the amount of interest m during a year. This means that the base period is a year divided by m, and the compound interest rate for the period is r/ m. The formula for compound interest, taking into account the signs of the Excel financial functions, will take the form: S+P( 1+ r/ m) t = 0. Parameter t measured in periods. If the calculation is k years, the formula takes the form S+P( 1+r/ m) km =0.
In addition to time-fixed interest rates, "floating" rates
(floating rate). The amount of accumulation with variable rates is determined by the formula: , where 
– successive values of interest rates in time; 
– periods of validity of the corresponding rates.
Example 2.2. The loan was issued for 5 years. The fixed part of the interest rate is set at 12% per annum plus surcharges (margin) of 0.5% in the first two years and 0.75% in the rest. Find the growth factor.
The multiplier will be:
q= (1+0.125) 2 (1+0.1275) 3 =1.81407
Often the interest period is not an integer number of years. In this case, two methods are used for calculation. With the general method, the calculation is carried out according to the compound interest formula. With the mixed method for an integer number of years, interest is calculated according to the compound interest formula, and for the fractional part of the period - according to the simple interest formula: 
, where a+
b=
t;
a
–
integer number of periods; b- fractional part of the period t.
To calculate compound interest problems, we use the same operation algorithm and financial functions as for simple interest.
In cell B1 we place the value of the initial value of the contribution. In cells B2: G2 we will place the numbers 0, 1, ..., 5, in cells AZ: A7 the values \u200b\u200bof 10%, 20%, ..., 50% (these numbers are entered using the methods of generating arithmetic progressions). It is necessary to tabulate the function of two variables (interest rate and number of years), depending on the parameter - the initial contribution. Let's enter the formula =BS ($AZ, B$2, -$B$1) into the cell OT. The formula is copied to the rest of the cells in the interval B3:G7.
Example 2.4. A loan of $ 20,000 was given for a year and a half at a rate of 28% per annum with a quarterly accrual. Determine the amount of the final payment.
Here the base period is a quarter. The term of the loan is 6 periods (4 quarters in a year, a period of one and a half years), 7% = 28% / 4 are charged for the period. Then the formula that gives the solution to the problem is: = BS (28% / 4, 4 * 1.5, 20000). It returns the result -$30014.61.
Tasks
4. The bank accepts a deposit for a period of 3 months with an announced annual rate of 100% or for 6 months at 110%. How profitable is it to invest money for six months: twice for three months or once for 6 months?
5. Amount 2000 rub. placed at 9% per annum for 3 years. Interest is calculated quarterly. What amount will be in the account?
6. What is the amount of debt after 26 months, if its initial value is $ 500,000, interest is compound, the rate is 20% per annum, accrual is quarterly? Carry out calculations by general and mixed methods.
7. The bank received a loan in the amount of 250 million rubles. The annual interest rate is 9.5% with an assumed year length of 360 days. Calculate the amount of accumulated debt by general and mixed methods for a different lending period, the duration of which:
equal to an integer number of years (without a fractional part) - 3 years;
equal to one year;
is less than a year - 0.25 years;
equals an integer number of years + a fraction of a year - 2 years and 270 days.
Compare the obtained values by options and identify patterns in the difference in the results.
